Subthreshold phi-meson production in heavy-ion collisions

17 Pages·0.17 MB·

by  H. W. Barz

Subthreshold φ meson production in
heavy-ion collisions
H.W. Barza, M. Z´et´enyib, Gy. Wolfb, B. K¨ampfera
aForschungszentrum Rossendorf, Pf 510119, 01314 Dresden, Germany
2 bKFKI Research Institute for Particle and Nuclear Physics, POB 49, H-1525
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Budapest, Hungary
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Abstract
J

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Within a transport code of BUU type the production of φ mesons in the reactions

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Ni + Ni at 1.93 AGeV and Ru + Ru at 1.69 AGeV is studied. New elementary
v · ·
3 reaction channels ρN(∆) φN and πN(1520) φN are included. In spite of a
→ →
1 substantial increase of the φ multiplicities by these channels the results stay below
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the tentative numbers extracted from experimental data.
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0 Key words: phi-meson, subthreshold production, heavy-ion collisions
/ PACS: 25.75.-q, 25.75.Dw
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1 Introduction
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Meson production in heavy-ion collisions is considered a sensitive probe of
the reaction dynamics [1]. Particularly interesting is the production of heavy
mesons at such a kinetic energy which is insufficient to create the respective
meson in a free, single nucleon-nucleon collision. Secondary and cooperative
processes are needed to accumulate the energy for the creation of the heavy
meson. Therefore, thestudy of thesubthreshold particleproductionisthought
to reveal many aspects of the behaviour of strongly interacting matter includ-
ing the equation of state [2].
Focusing on heavy-ion collisions at bombarding energies in the range of one
A GeV the production of strangeness degrees of freedom attracted much at-
·
Email addresses: [email protected](H.W. Barz),
[email protected] (M. Z´et´enyi), [email protected] (Gy. Wolf),
[email protected] (B. Ka¨mpfer).
Preprint submitted to Elsevier Science 8 February 2008
tention, both experimentally and theoretically. For instance, the recent mea-
surements seem to comply with a noticeable mass splitting between the K−
and K+ mesons [3], as predicted by various effective (chiral) models for pseu-
doscalar mesons [4].
In contrast to the comparatively rich body of experimental data on open
strangenessmesonproduction,theinformationonmesonswithhiddenstrange-
ness, i.e. the φ, is scarce. Only a few measurements [5,6] for the reactions Ni
+ Ni at 1.93 A GeV and Ru + Ru at 1.69 A GeV have been reported. In these
· ·
measurements a strongly limited amount of the phase space was accessible.
Extrapolations to the full phase space point to a surprisingly large production
cross section. The results of the analysis [7] for the reaction Ni + Ni [5] let one
recognise that the calculations based on a transport model underestimate the
data by more than a factor of ten. (Notice that refined re-analyses [6,8] of this
reactionhasshiftedthepreviously reportedφyield [5]toalargervalue.) Inthe
calculations [7] the φ mesons are solely produced in the elementary processes
BB φBB and πB φB, where the symbol B denotes the baryons N and
→ →
∆. Very recently the analysis of the φ production within another transport
model has been repeated [9] with the same production channels confirming
the previous finding [7]. The discrepancy of these calculations with the data
[6,8,10] has inspired us to study whether further elementary channels might
essentially contribute to the φ meson production.
In Ref. [11] the role of three-nucleon collisions for φ production was investi-
gated. It turned out that these processes are well described by a sequence of
two two-body processes inside the dense medium. However, to obtain the cor-
rect yield, one has to include all intermediate particles which can be created
inside the medium. This concerns especially the ρ mesons which have a rather
large production cross section and a low threshold energy when colliding with
baryons.
Furthermore, even at subthreshold bombarding energies of about 1.5 up to 2
GeV per nucleon, up to 30% of the nucleons are excited to ∆ and N∗ reso-
nances which have a lower thresholdforφmeson productionthangroundstate
nucleons when colliding with other particles. It is the aim of this study to in-
vestigate the effects of these additional production channels which have not
been studied so far. To be specific, we consider here the elementary reactions
ρB Bφ and πN(1520) Nφ.
→ →
The study of the φ meson production is also of interest concerning the K−
production. Due to the strong decay channel φ K+K− a substantial feed-
→
ing of the K− channel could not be ruled out. This is particularly tempting
as the analysis [6] points to a substantial fraction of K− stemming from φ
decays. In the near future also the decay channel φ e+e− becomes exper-
→
imentally accessible with the HADES detector at the heavy-ion synchrotron
2
SIS at GSI/Darmstadt [12]. Due to the large ss¯ quark content, φ mesons are
expected to interact much weaker with the non-strange baryonic matter in
accordance with the Okubo-Zweig-Iizuka rule (cf. [13]). Nevertheless, notice-
able in-medium effects, e.g. mass shift and broadening, are expected at higher
temperature and density [14,15].
Our paper is organised as follows. In section 2 the cross sections of the above
mentioned elementary hadron reactions are estimated within a one-boson ex-
change model. Sections 3 and 4 describe the resulting φ multiplicities when
implementing these new reaction channels in a transport code. In section 5 we
compare with available experimental results. The summary and the discussion
can be found in section 6.
2 Elementary cross sections
We are going to describe heavy-ion collisions within the Boltzmann-Uehling-
Uhlenbeck (BUU) formalism. At beam energies around 2 GeV per nucleon
many resonances are generated. Therefore, there are many possibilities to pro-
duce φ mesons by two-body collisions. In Refs. [7,9] the cross sections of the
collisions of the most abundant particles, NN NN φ, N∆ NNφ, ∆∆
→ → →
NN φ, πN Nφ, and π∆ Nφ, have been used. Since the cross sections
→ →
are not well known a one-boson exchange model has been employed to es-
timate them [7,16]. The parameters of the corresponding Lagrangians have
been fixed by decay widths and cross sections measured at different energies.
Selected new datanear threshold becamerecently available [17].Asmentioned
above, employing these cross sections within transport codes one gets φ meson
production rates which seriously underestimate the extrapolations made from
the measurements [5,6].
Channels with ρ mesons as incoming particles have not been included in
Refs. [7,9]. Since the φ meson decays into the non-strange sector overwhelm-
ingly via the πρ channel (with 13% branching ratio, see [18]), both mesons are
expected to play an important role for the φ production. Other channels are
substantially weaker.
2.1 φ meson production in ρ-baryon collisions
Let us consider first the φ production in ρ induced reactions on the nucleon
and the ∆, i.e. ρN φN and ρ∆ φN. Figure 1a shows the correspond-
→ →
ing tree level diagram hitherto not yet considered. We calculate the cross
sections within the one-boson exchange model with the effective interaction
3
N φ N φ φ
π ρ
N,∆ ρ N* π π ρ
a) b) c)
Fig. 1. Diagrammatic representation of processes contributing to the reactions (a)
ρ (N,∆) φ N, (b) π N∗ φ N and (c) πρ φ.
→ → →
Lagrangians
f
= πNNψ¯γ γµ~τψ ∂ ~π, (1)
πNN 5 µ
L m ·
π
f
= πN∆ψ¯T~ψµ ∂ ~π + h.c., (2)
πN∆ µ
L − m ·
π
f
= πρφǫ ∂µφν∂αρ~β ~π. (3)
πρφ µναβ
L m ·
φ
Here ψ is the nucleon field, ψ is the Rarita-Schwinger field for the ∆ reso-
µ
nance, and π, ρ~ , φ are the meson fields while m and m stand for the pion
µ µ π φ
and ρ meson masses, respectively. The symbol ~τ denotes the isospin Pauli
matrix, and T~ is the isospin 1/2 3/2 transition matrix. The diagrams in
→
Fig. 1 are to be understood as guidance to construct Lorentz invariant ampli-
tudes from effective interaction Lagrangians. They must be supplemented by
additional elements to achieve a reasonable description of elementary hadron
processes, for which data are available. In this line of arguments we intro-
duce multipole form factors for off-shell pions with four-momentum p at each
π
vertex which, e.g. for the πNN vertex reads
n
(Λπ )2 m2
Fπ (p2) = πNN − π (4)
πNN π  (Λπ )2 p2 
πNN − π
 
with n = 1, while for ρ mesons we use a corresponding expression with n = 2.
The cross section for the ρ∆ φN reaction is regularised by the Peierls
→
method [19]. We adopted the coupling constants f and cutoff parameters Λ
i i
for the π NN and πN∆ vertices from the Model II in Table B.1 of Ref. [20].
For f and Λπ we used the values given in [7].
πρφ πρφ
4
n p ∆− ∆0 ∆+ ∆++
ρ− 0 2 0 0 1 3
ρ0 1 1 0 2 2 0
ρ+ 2 0 3 1 0 0
Table 1
Isospin factors for the various channels of the reactions ρN φN and ρ∆ φN.
→ →
The isospin averaged differential cross sections can be written as
dσ 1 p
= | out,CM| 2, (5)
dΩ 64π2s p |M|
in,CM
| |
with s being the square of the center-of-mass energy and p the incom-
in(out),CM
ing (outgoing) three-momentum in the center-of-mass system. The quantity
2 represents the squared matrix elements summed over the final and av-
|M|
eraged over the initial spin and isospin states. These quantities are given by
2
16 A
|MρN→φN|2 = 3 p2πρN−→mφN2π! m2N (cid:16)pN ·pN′ −m2N(cid:17)(cid:16)(pρ ·pφ)2 −m2ρm2φ(cid:17)
(6)
for the ρN channel and
2
4 A
2 = ρ∆→φN
|Mρ∆→φN| 27 p2π −m2π!
p p 2
(p p +m m ) ∆ · N m2 (p p )2 m2m2 (7)
× ∆ · N ∆ N (cid:18) m∆ (cid:19) − N!(cid:16) ρ · φ − ρ φ(cid:17)
for the ρ∆ channel. Here p , p , p , and p denote the four-momenta of
N ∆ ρ φ
the respective particles. In the case of the ρN channel, p and p′ refer to
N N
the four-momenta of the incoming and outgoing nucleon, respectively. The
factors A are given by A = (f f /m m )Fπ (p2)Fπ (p2) and
i ρN→φN πNN πρφ π φ πNN π πρφ π
A = (f f /m m )Fπ (p2)Fπ (p2). The total cross sections are
ρ∆→φN πN∆ πρφ π φ πN∆ π πρφ π
obtained by numerically integrating Eq. (5) over the solid angle Ω.
Cross sections of the specific isospin channels can be deduced from the isospin
averagedcrosssectionbymultiplyingwithanappropriateisospinfactor.These
isospinfactors,whichcanbeexpressedintermsofClebsch-Gordancoefficients,
are given in Table 1.
2.2 φ formation from the πρ channel
The cross section of the πρ process has a very narrow resonance structure with
a width of Γ = 4.43 MeV and a branching ratio of Γ /Γ = 0.13 in the
tot πρ tot
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πρ channel. The formation cross section, depicted in Fig. 1c, of the φ meson
in the π+ρ− or the π−ρ+ channel is correspondingly
π Γ Γ
πρ tot
σ = (8)
2p2 (√s m )2 +Γ2 /4
in φ tot
−
which gives a maximum cross section of 9.5 mb at resonance peak position.
The π0ρ0 channel is isospin forbidden.
2.3 Effects of higher resonances
At bombarding energies of about 2 GeV per nucleon baryons can be highly
excited. Consequently the production threshold is reduced by their excitation
energy if φ mesons are created in collisions with other particles. To estimate
the importance of such processes we test the influence of the collision of a
π meson with the N(1520) 3/2− resonance. The corresponding diagram is
exhibited in Fig. 1b. This resonance is chosen because it decays with a large
branching ratio of about (20 5)% [18,21] into the ρ channel, although this
±
channel would be closed for the nominal masses of the ρ and N(1520). This
hints to the fact that a large part of the production cross section proceeds via
virtual ρ propagation as described by the diagram in Fig. 1b. This assumption
may not be applicable to higher resonances which can emit ρ mesons nearly
on shell.
There are several possibilities to describe the ρNN(1520) vertex by using sim-
ple effective interaction Lagrangians:
=g ψ¯ ~τψµ ρ~ + h.c., (9)
LρNN(1520) 0 N 1520 µ
f
= 1 ψ¯ ~τγνψµ (∂ ρ~ ∂ ρ~ ) + h.c., (10)
LρNN(1520) m N 1520 ν µ − µ ν
ρ
f
= 2 ψ¯ ~τσαβψµ ∂ ∂ ρ~ + h.c. . (11)
LρNN(1520) m2 N 1520 α µ β
ρ
The second Lagrangian is related to the vector dominance approach (see e.g.
[22]) while the last one follows from constituent-quark models [23]. We fix the
coupling constants by assuming that each Lagrangian alone would provide
the partial width to the ρ(two-pion) decay channel. Such a procedure supplies
upper limits on the coupling constants. We obtain the values g = 7.5, f =
0 1
10.5 and f = 68, respectively. With these values the Lagrangians (9) and
2
(10) give roughly the same cross section near threshold. Our coefficient f
1
is larger than the corresponding coefficient in Ref. [22], which is caused by
the monopole form factor used there. In Fig. 2 we present the cross sections
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4
10
π+ρ
103 lagrangian (11)
N1520
lagrangian (10)
2
10
) ρ+∆ ρ+N
b
µ
( π+N
σ
10
π+N1520
1 π+∆ ∆+∆
N+N
-1 N+∆
10
0 0.2 0.4 0.6 0.8 1 1.2 1.4
kinetic energy (GeV)
Fig. 2. Production cross sections for φ mesons for several binary reactions as a
function of the relative kinetic centre-of-mass energy in the entrance channel. The
cross sections are calculated for the nominal masses of the resonances. Especially
cross sections for ρ∆ collisions having different masses can considerably deviate
from the curve shown.
obtained with the Lagrangians (10) and (11). The large coupling constant
f needed for the Lagrangian (11) leads to a much larger cross section for
2
the φ production than the Lagrangian (10). This value of f , however, differs
2
from the value estimated from the quark model [23]. As our Lagrangians are
nothing more than a covariant parametrisation of the interaction the coupling
constants are generally energy dependent. The larger relevant energy in φ
productioncomparedtotheN(1520)decaycouldbethereasonforthedifferent
estimated values of the coupling constants. In the following we use the cross
section from the Lagrangian (10) as standard in our BUU model calculations.
We also performed calculations using the cross section calculated with the
Lagrangian (11) in order to estimate the uncertainties of the φ meson yield
from the π + N(1520) channel, see Fig. 2.
7
2.4 Summary of cross sections
Summarising the results of this section we show the calculated individual cross
sections in Fig. 2 as a function of the kinetic energy of the relative motion
in the entrance channel. Obviously, the ρ B channels are very important at
low energies. We also studied the reaction ππ φπ which, however, is not
→
important because of the large threshold energy which can rarely be overcome
by the pions. For the φ production via the BB and πB channels we use the
same cross sections as found in Ref. [7].
3 BUU calculations
To calculate φ meson production in a heavy-ion reaction we use the version of
the BUU code introduced in Ref. [24]. The production of φ mesons proceeds
via the reactions discussed in the previous section. To increase the statistics
the perturbative method is used, i.e. if in a two-particle collision the threshold
is overcome, the φ meson is created and weighted with its production proba-
bility. However the production process itself and its subsequent collisions do
not influence the motion of the other participating particles. To simplify the
calculations an isotropic cross section is assumed for each elementary hadron
reaction and in the case of a three particle final state the energy distribution
is taken from the phase space.
In the transport model all nucleon and delta resonances are included up to 2.2
GeV. Furthermore, Λ and Σ baryons as well as the mesons π,η,ρ,σ,ω, K+
and K0 are explicitly propagated. The properties (masses, widths, branching
ratios) of the baryon resonances have been fitted [25] to the one- and two-
meson production data available from pion-nucleon reactions. The production
cross sections for resonances in nucleon-nucleon collision arederived fromdata
on meson production in nucleon-nucleon reactions.
Furthermore, momentum dependent potentials for the nucleons are included.
The same potentials are also used for the ∆ particles. As standard potential
we employ such one which delivers a soft equation of state with an incom-
pressibility parameter of 215 MeV. Less is known about in-medium properties
of the φ mesons. While in Ref. [15] a very small mass shift has been found,
we use for exploratory purposes the prediction by Hatsuda et al. [14] for the
density dependence of the φ meson mass:
n
m∗ = m 0.025m . (12)
φ φ − φn
0
8
0.5
ρ
0.4
r
e
b
m 0.3
u
n

e
l φ x100
c
0.2
i
t
r
a
p
φ x100
0.1
ρ(π+π)
0
0 5 10 15 20 25
time (fm/c)
Fig. 3. Number of produced ρ and φ mesons as a function of time for the reaction
Ru on Ru at beam energy of 1.69 AGeV and impact parameter b = 2.8 fm. The
·
lower pair of curves is calculated by switching off the ρ production dueto resonance
decays.
The second term is treated as the mean field of the φ meson.
The interaction of the φ mesons with surrounding nucleons is relatively weak.
The elastic cross section is estimated to be in the order of 0.5 mb, however, a
strong absorption cross section is expected via the reaction channel φB KΛ.
→
Following Ref. [26] we use a value of 6mb exp( 3.3ǫ/GeV) with ǫ being the
· −
kinetic energy in the entrance channel. Similar values are employed in Ref. [7].
Using these potentials we have calculated the φ production cross sections for
thereactionsNionNiat1.93A GeVandRuonRuat1.69A GeVbombarding
· ·
energy. The results are reported in Table 2.
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yields from Ni + Ni (1.93 GeV) Ru + Ru (1.69 GeV)
B + B 3.5 10−4 3.1 10−4
· ·
π + B 2.9 10−4 3.2 10−4
· ·
ρ + B 8.9 10−4 11.8 10−4
· ·
π + ρ 1.6 10−4 1.5 10−4
· ·
π + N(1520) 0.5 10−4 0.6 10−4
· ·
total yield 1.7 10−3 2.0 10−3
· ·
in HELITRON 1.2 10−5 1.5 10−5
· ·
in CDC 2.7 10−5 2.9 10−5
· ·
experiment [6] (8.7 3.6) 10−3 (6.4 2.5) 10−3
± · ± ·
Table 2
Multiplicities of φ mesons per central event in Ni + Ni and Ru + Ru reactions.
The first 5 lines give the results from the special channels indicated. The symbol B
comprises the nucleon and the ∆ particle. Lines 7 and 8 are the yields observed in
the two detector subsystems.
4 Influence of model parameters
In the first five lines of Table 2 the contributions of the different production
channels are listed. We notice a substantial increase by a factor of 2.5 of the φ
multiplicities when including the new ρ channels. The π + N(1520) reaction
channel calculated with the Lagrangian (10) does not essentially contribute
to the φ production. However, when using the cross sections obtained from
the Lagrangian (11) the φ meson yield from the π + N(1520) channel rises
by more than an order of magnitude to 6.6 10−4 (7.7 10−4) for the reaction Ni
· ·
+ Ni (Ru + Ru). This indicates that the π + N(1520) channel might not be
negligible for the φ production, but to clarify this requires a better knowledge
of the ρNN(1520) vertex.
Inordertotestthedependence oftheφmesonyieldontheequationofstatewe
carried out also calculations with different potentials. For instance, employing
a hard momentum dependent potential with an incompressibility parameter
of 380 MeV only two third of the φ yield of the soft equation of state has
been obtained for the Ru on Ru reaction. The reduction of the φ multiplicity
concerns especially the BB and πB channels while the ρB and πρ channels
are less influenced. Obviously, to overcome the larger potentials the collision
partners use up a large amount of kinetic energy which then lacks for the φ
production. Similar effects are known for kaon production, see Ref. [3]. We
have also found that momentum independent potentials enhance the number
of the produced φ mesons.
10